Pair of Opposite Sides of a Parallelogram are Equal and Parallel

Here we will discuss about one of the important geometrical property of parallelogram.

A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel

Given: PQRS is a quadrilateral in which PQ = SR and PQ ∥ SR.

To prove: PQRS is a parallelogram.

Construction: Join PR and QS such that they intersect at O.

Pair of Opposite Sides of a Parallelogram are Equal and Parallel




In ∆OPQ and ∆ORS,

1. ∠OPQ = ∠ORS

1. PQ ∥ SR and PR is a transversal.

2. ∠POQ = ∠ROS

2. Opposite angles are equal.

3. PQ = RS

3. Given.

4. ∆OPQ ≅ ∆ORS

Therefore, OP = OR, OQ = OS.

In ∆OPS and ∆OQR,

4. By AAS criterion of congruency.


5. OP = OC, OQ = OS, ∠POS = ∠QOR

5. By statement 4 and reason 2.

6. ∆OPS ≅ ∆OQR

Therefore, PS = QR, ∠OPS= ∠ORQ

6. By SAS criterion of congruency.


7. PS ∥QR.

7. Alternate angles are equal.

8. PQRS is a parallelogram (Proved).

8. PQ ∥ SR and statement 7.

Corollary: In a parallelogram, each pair of opposite sides are parallel as well as equal.

9th Grade Math

From Pair of Opposite Sides of a Parallelogram are Equal and Parallel to HOME PAGE

New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Share this page: What’s this?

Recent Articles

  1. Fraction as a Part of Collection | Pictures of Fraction | Fractional

    Feb 24, 24 04:33 PM

    Pictures of Fraction
    How to find fraction as a part of collection? Let there be 14 rectangles forming a box or rectangle. Thus, it can be said that there is a collection of 14 rectangles, 2 rectangles in each row. If it i…

    Read More

  2. Fraction of a Whole Numbers | Fractional Number |Examples with Picture

    Feb 24, 24 04:11 PM

    A Collection of Apples
    Fraction of a whole numbers are explained here with 4 following examples. There are three shapes: (a) circle-shape (b) rectangle-shape and (c) square-shape. Each one is divided into 4 equal parts. One…

    Read More

  3. Identification of the Parts of a Fraction | Fractional Numbers | Parts

    Feb 24, 24 04:10 PM

    Fractional Parts
    We will discuss here about the identification of the parts of a fraction. We know fraction means part of something. Fraction tells us, into how many parts a whole has been

    Read More

  4. Numerator and Denominator of a Fraction | Numerator of the Fraction

    Feb 24, 24 04:09 PM

    What are the numerator and denominator of a fraction? We have already learnt that a fraction is written with two numbers arranged one over the other and separated by a line.

    Read More

  5. Roman Numerals | System of Numbers | Symbol of Roman Numerals |Numbers

    Feb 24, 24 10:59 AM

    List of Roman Numerals Chart
    How to read and write roman numerals? Hundreds of year ago, the Romans had a system of numbers which had only seven symbols. Each symbol had a different value and there was no symbol for 0. The symbol…

    Read More